Existence and Numerical Approximation of Solutions to Nonlinear Fractional Differential Equations in Banach Spaces

Abstract

Nonlinear fractional differential equations (FDEs) have become strong modeling methods of complex dynamical systems and memory and hereditary phenomena, especially in infinite dimensional functional spaces. We explore the existence and numerical approximation of solutions to a family of nonlinear fractional differential equations, formulated in Banach spaces, in this study. Theoretical solution is performed by transforming the problem into an equivalent integral equation and using fixed-point methods, such as the Banach contraction principle and Schauder fixed-point theorem to prove the existence (and uniqueness when appropriate) of solutions. At the computational level a fractional Adams-Bashforth-Moulton predictor-corrector scheme is used to get numerical approximations of the solution. The consistency of the proposed numerical framework, its stability, and convergence are strictly examined. Estimates of errors are obtained based on well-known norms, such as the L2 and the L-infinity norms, which prove that the numerical solution approaches the exact solution as the step size is reduced. The theoretical results are verified by numerical experiments on representative models of nonlinear FDEs which are highly accurate and converge better than the classical approximation techniques. The findings support the claim that the suggested methodology is effective to bridge between the theoretical analysis and computation. In general, the work offers the detailed framework incorporating the existence theory and sound numerical approximation methods to present the scalable approach to solving nonlinear fractional system in Banach spaces to a mathematically consistent solution.